Integrand size = 35, antiderivative size = 293 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {(2 A-7 B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{7/2} d}-\frac {(177 A-637 B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {(A-B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(3 A-7 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}+\frac {(79 A-259 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {7 (7 A-27 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}} \] Output:
(2*A-7*B)*arcsin(a^(1/2)*sin(d*x+c)/(a+a*cos(d*x+c))^(1/2))/a^(7/2)/d-1/12 8*(177*A-637*B)*arctan(1/2*a^(1/2)*sin(d*x+c)*2^(1/2)/cos(d*x+c)^(1/2)/(a+ a*cos(d*x+c))^(1/2))*2^(1/2)/a^(7/2)/d+1/6*(A-B)*cos(d*x+c)^(7/2)*sin(d*x+ c)/d/(a+a*cos(d*x+c))^(7/2)+1/16*(3*A-7*B)*cos(d*x+c)^(5/2)*sin(d*x+c)/a/d /(a+a*cos(d*x+c))^(5/2)+1/192*(79*A-259*B)*cos(d*x+c)^(3/2)*sin(d*x+c)/a^2 /d/(a+a*cos(d*x+c))^(3/2)-7/64*(7*A-27*B)*cos(d*x+c)^(1/2)*sin(d*x+c)/a^3/ d/(a+a*cos(d*x+c))^(1/2)
Time = 5.13 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.29 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {a (1+\cos (c+d x))} \left (-336 (7 A-27 B) \arcsin \left (\sqrt {1-\cos (c+d x)}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )-8496 A \arcsin \left (\sqrt {\cos (c+d x)}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )+30576 B \arcsin \left (\sqrt {\cos (c+d x)}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )+24 \sqrt {2} (177 A-637 B) \arctan \left (\frac {\sqrt {\cos (c+d x)}}{\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )-724 A \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)+2884 B \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)-494 A \sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)+2198 B \sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)+384 B \sqrt {1-\cos (c+d x)} \cos ^{\frac {7}{2}}(c+d x)-294 A \sqrt {-((-1+\cos (c+d x)) \cos (c+d x))}+1134 B \sqrt {-((-1+\cos (c+d x)) \cos (c+d x))}\right ) \sin (c+d x)}{384 a^4 d \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^4} \] Input:
Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^( 7/2),x]
Output:
(Sqrt[a*(1 + Cos[c + d*x])]*(-336*(7*A - 27*B)*ArcSin[Sqrt[1 - Cos[c + d*x ]]]*Cos[(c + d*x)/2]^6 - 8496*A*ArcSin[Sqrt[Cos[c + d*x]]]*Cos[(c + d*x)/2 ]^6 + 30576*B*ArcSin[Sqrt[Cos[c + d*x]]]*Cos[(c + d*x)/2]^6 + 24*Sqrt[2]*( 177*A - 637*B)*ArcTan[Sqrt[Cos[c + d*x]]/Sqrt[Sin[(c + d*x)/2]^2]]*Cos[(c + d*x)/2]^6 - 724*A*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2) + 2884*B*Sqr t[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2) - 494*A*Sqrt[1 - Cos[c + d*x]]*Cos[ c + d*x]^(5/2) + 2198*B*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(5/2) + 384*B* Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(7/2) - 294*A*Sqrt[-((-1 + Cos[c + d*x ])*Cos[c + d*x])] + 1134*B*Sqrt[-((-1 + Cos[c + d*x])*Cos[c + d*x])])*Sin[ c + d*x])/(384*a^4*d*Sqrt[1 - Cos[c + d*x]]*(1 + Cos[c + d*x])^4)
Time = 1.99 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.08, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.543, Rules used = {3042, 3456, 27, 3042, 3456, 27, 3042, 3456, 27, 3042, 3462, 25, 3042, 3461, 3042, 3253, 223, 3261, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a \cos (c+d x)+a)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) (7 a (A-B)-2 a (A-7 B) \cos (c+d x))}{2 (\cos (c+d x) a+a)^{5/2}}dx}{6 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) (7 a (A-B)-2 a (A-7 B) \cos (c+d x))}{(\cos (c+d x) a+a)^{5/2}}dx}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (7 a (A-B)-2 a (A-7 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}dx}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (15 a^2 (3 A-7 B)-2 a^2 (17 A-77 B) \cos (c+d x)\right )}{2 (\cos (c+d x) a+a)^{3/2}}dx}{4 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (15 a^2 (3 A-7 B)-2 a^2 (17 A-77 B) \cos (c+d x)\right )}{(\cos (c+d x) a+a)^{3/2}}dx}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (15 a^2 (3 A-7 B)-2 a^2 (17 A-77 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 \sqrt {\cos (c+d x)} \left (a^3 (79 A-259 B)-14 a^3 (7 A-27 B) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x) a+a}}dx}{2 a^2}+\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {\sqrt {\cos (c+d x)} \left (a^3 (79 A-259 B)-14 a^3 (7 A-27 B) \cos (c+d x)\right )}{\sqrt {\cos (c+d x) a+a}}dx}{4 a^2}+\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a^3 (79 A-259 B)-14 a^3 (7 A-27 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}+\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3462 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int -\frac {7 a^4 (7 A-27 B)-64 a^4 (2 A-7 B) \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{a}-\frac {14 a^3 (7 A-27 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )}{4 a^2}+\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\int \frac {7 a^4 (7 A-27 B)-64 a^4 (2 A-7 B) \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{a}-\frac {14 a^3 (7 A-27 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )}{4 a^2}+\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\int \frac {7 a^4 (7 A-27 B)-64 a^4 (2 A-7 B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{a}-\frac {14 a^3 (7 A-27 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )}{4 a^2}+\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3461 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {a^4 (177 A-637 B) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx-64 a^3 (2 A-7 B) \int \frac {\sqrt {\cos (c+d x) a+a}}{\sqrt {\cos (c+d x)}}dx}{a}-\frac {14 a^3 (7 A-27 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )}{4 a^2}+\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {a^4 (177 A-637 B) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-64 a^3 (2 A-7 B) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}-\frac {14 a^3 (7 A-27 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )}{4 a^2}+\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3253 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {a^4 (177 A-637 B) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx+\frac {128 a^3 (2 A-7 B) \int \frac {1}{\sqrt {1-\frac {a \sin ^2(c+d x)}{\cos (c+d x) a+a}}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}}{a}-\frac {14 a^3 (7 A-27 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )}{4 a^2}+\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {a^4 (177 A-637 B) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {128 a^{7/2} (2 A-7 B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{a}-\frac {14 a^3 (7 A-27 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )}{4 a^2}+\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 3261 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {2 a^5 (177 A-637 B) \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x) a^3}{\cos (c+d x) a+a}+2 a^2}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}\right )}{d}-\frac {128 a^{7/2} (2 A-7 B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{a}-\frac {14 a^3 (7 A-27 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )}{4 a^2}+\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {a^2 (79 A-259 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac {3 \left (-\frac {\frac {\sqrt {2} a^{7/2} (177 A-637 B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{d}-\frac {128 a^{7/2} (2 A-7 B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{a}-\frac {14 a^3 (7 A-27 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )}{4 a^2}}{8 a^2}+\frac {3 a (3 A-7 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\) |
Input:
Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(7/2),x ]
Output:
((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((3*a*(3*A - 7*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((a^2*(79*A - 259*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d* (a + a*Cos[c + d*x])^(3/2)) + (3*(-(((-128*a^(7/2)*(2*A - 7*B)*ArcSin[(Sqr t[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (Sqrt[2]*a^(7/2)*(177*A - 637*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d)/a) - (14*a^3*(7*A - 27*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])))/(4*a^2))/(8*a^2))/(12*a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[-2/f Subst[Int[1/Sqrt[1 - x^2/a], x], x, b*(Co s[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, d, e, f}, x] && E qQ[a^2 - b^2, 0] && EqQ[d, a/b]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f) Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*S in[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] & & NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[c, 0])
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim p[(A*b - a*B)/b Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) , x], x] + Simp[B/b Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]] , x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(b*(m + n + 1)) Int[(a + b*Sin[e + f*x])^m*(c + d*S in[e + f*x])^(n - 1)*Simp[A*b*c*(m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Time = 17.03 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.44
| method | result | size |
| default | \(\frac {\left (\left (384 \cos \left (d x +c \right )^{3}+1152 \cos \left (d x +c \right )^{2}+1152 \cos \left (d x +c \right )+384\right ) \sqrt {2}\, A \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+\left (-1344 \cos \left (d x +c \right )^{3}-4032 \cos \left (d x +c \right )^{2}-4032 \cos \left (d x +c \right )-1344\right ) \sqrt {2}\, B \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+\sin \left (d x +c \right ) \left (-247 \cos \left (d x +c \right )^{2}-362 \cos \left (d x +c \right )-147\right ) \sqrt {2}\, A \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\sin \left (d x +c \right ) \left (192 \cos \left (d x +c \right )^{3}+1099 \cos \left (d x +c \right )^{2}+1442 \cos \left (d x +c \right )+567\right ) \sqrt {2}\, B \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (531 \cos \left (d x +c \right )^{3}+1593 \cos \left (d x +c \right )^{2}+1593 \cos \left (d x +c \right )+531\right ) A \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+\left (-1911 \cos \left (d x +c \right )^{3}-5733 \cos \left (d x +c \right )^{2}-5733 \cos \left (d x +c \right )-1911\right ) B \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\cos \left (d x +c \right )}}{192 d \left (\cos \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, a^{4}}\) | \(422\) |
| parts | \(\frac {A \left (\left (384 \cos \left (d x +c \right )^{3}+1152 \cos \left (d x +c \right )^{2}+1152 \cos \left (d x +c \right )+384\right ) \sqrt {2}\, \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+\sin \left (d x +c \right ) \left (-247 \cos \left (d x +c \right )^{2}-362 \cos \left (d x +c \right )-147\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (531 \cos \left (d x +c \right )^{3}+1593 \cos \left (d x +c \right )^{2}+1593 \cos \left (d x +c \right )+531\right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{192 d \left (\cos \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, a^{4}}+\frac {B \left (\left (-1344 \cos \left (d x +c \right )^{3}-4032 \cos \left (d x +c \right )^{2}-4032 \cos \left (d x +c \right )-1344\right ) \sqrt {2}\, \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+\sin \left (d x +c \right ) \left (192 \cos \left (d x +c \right )^{3}+1099 \cos \left (d x +c \right )^{2}+1442 \cos \left (d x +c \right )+567\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (-1911 \cos \left (d x +c \right )^{3}-5733 \cos \left (d x +c \right )^{2}-5733 \cos \left (d x +c \right )-1911\right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{192 d \left (\cos \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, a^{4}}\) | \(510\) |
Input:
int(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^(7/2),x,method=_RET URNVERBOSE)
Output:
1/192/d*((384*cos(d*x+c)^3+1152*cos(d*x+c)^2+1152*cos(d*x+c)+384)*2^(1/2)* A*arctan(tan(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+(-1344*cos(d*x+c)^3 -4032*cos(d*x+c)^2-4032*cos(d*x+c)-1344)*2^(1/2)*B*arctan(tan(d*x+c)*(cos( d*x+c)/(cos(d*x+c)+1))^(1/2))+sin(d*x+c)*(-247*cos(d*x+c)^2-362*cos(d*x+c) -147)*2^(1/2)*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+sin(d*x+c)*(192*cos(d*x+ c)^3+1099*cos(d*x+c)^2+1442*cos(d*x+c)+567)*2^(1/2)*B*(cos(d*x+c)/(cos(d*x +c)+1))^(1/2)+(531*cos(d*x+c)^3+1593*cos(d*x+c)^2+1593*cos(d*x+c)+531)*A*a rcsin(cot(d*x+c)-csc(d*x+c))+(-1911*cos(d*x+c)^3-5733*cos(d*x+c)^2-5733*co s(d*x+c)-1911)*B*arcsin(cot(d*x+c)-csc(d*x+c)))*(a*cos(1/2*d*x+1/2*c)^2)^( 1/2)*cos(d*x+c)^(1/2)/(cos(d*x+c)^4+4*cos(d*x+c)^3+6*cos(d*x+c)^2+4*cos(d* x+c)+1)/(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)/a^4
Time = 12.63 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.39 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=-\frac {3 \, \sqrt {2} {\left ({\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (177 \, A - 637 \, B\right )} \cos \left (d x + c\right ) + 177 \, A - 637 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, {\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) - 2 \, {\left (192 \, B \cos \left (d x + c\right )^{3} - {\left (247 \, A - 1099 \, B\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (181 \, A - 721 \, B\right )} \cos \left (d x + c\right ) - 147 \, A + 567 \, B\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 384 \, {\left ({\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right ) + 2 \, A - 7 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )}\right )}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \] Input:
integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^(7/2),x, algo rithm="fricas")
Output:
-1/384*(3*sqrt(2)*((177*A - 637*B)*cos(d*x + c)^4 + 4*(177*A - 637*B)*cos( d*x + c)^3 + 6*(177*A - 637*B)*cos(d*x + c)^2 + 4*(177*A - 637*B)*cos(d*x + c) + 177*A - 637*B)*sqrt(a)*arctan(1/2*sqrt(2)*sqrt(a*cos(d*x + c) + a)* sqrt(a)*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x + c)^2 + a*cos(d*x + c) )) - 2*(192*B*cos(d*x + c)^3 - (247*A - 1099*B)*cos(d*x + c)^2 - 2*(181*A - 721*B)*cos(d*x + c) - 147*A + 567*B)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d *x + c))*sin(d*x + c) - 384*((2*A - 7*B)*cos(d*x + c)^4 + 4*(2*A - 7*B)*co s(d*x + c)^3 + 6*(2*A - 7*B)*cos(d*x + c)^2 + 4*(2*A - 7*B)*cos(d*x + c) + 2*A - 7*B)*sqrt(a)*arctan(sqrt(a*cos(d*x + c) + a)*sqrt(a)*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x + c)^2 + a*cos(d*x + c))))/(a^4*d*cos(d*x + c )^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c) + a^4*d)
Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(7/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))**(7/2),x)
Output:
Timed out
\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^(7/2),x, algo rithm="maxima")
Output:
integrate((B*cos(d*x + c) + A)*cos(d*x + c)^(7/2)/(a*cos(d*x + c) + a)^(7/ 2), x)
Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^(7/2),x, algo rithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{7/2}\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \] Input:
int((cos(c + d*x)^(7/2)*(A + B*cos(c + d*x)))/(a + a*cos(c + d*x))^(7/2),x )
Output:
int((cos(c + d*x)^(7/2)*(A + B*cos(c + d*x)))/(a + a*cos(c + d*x))^(7/2), x)
\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\cos \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+1}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+1}d x \right ) a \right )}{a^{4}} \] Input:
int(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^(7/2),x)
Output:
(sqrt(a)*(int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x)**4)/ (cos(c + d*x)**4 + 4*cos(c + d*x)**3 + 6*cos(c + d*x)**2 + 4*cos(c + d*x) + 1),x)*b + int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x)**3 )/(cos(c + d*x)**4 + 4*cos(c + d*x)**3 + 6*cos(c + d*x)**2 + 4*cos(c + d*x ) + 1),x)*a))/a**4